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<title>Simulations for Statistical and Thermal Physics</title>

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<h3 style="text-align:center;">Estimating the chemical potential using the Widom insertion method</h3>

<p class="header_title"><b>Introduction</b></p>

<p>For a system at a given temperature and volume the chemical potential is given by</p>
<p class="center">
&#956; = -kT ln(Z<sub>N+1</sub>/Z<sub>N</sub>),
</p>
<p>where Z<sub>N</sub> is the partition function for N particles. For a classical system the partition function contains an integral over the momentum degrees of freedom and an integral over the position degrees of freedom. The integral over the momentum gives the ideal gas contribution to the chemical potential, and the integral over the coordinates of the particles yields the excess chemical potential &#956;*. That is, the total chemical potential is given by &#956; = &#956;<sub>ideal</sub> + &#956;*. It is easy to see that the excess chemical potential can be expressed for sufficiently large N as</p>
<p class="center">
&#956;* = -kT ln&lt;exp(-&#916;U/kT)&gt;,
</p>
<p>where the average is computed by averaging over random potential additions of a particle. The potential energy &#916;U is the change in potential energy of the system that would be found if we were to actually add the particle. During the simulation no particle is actually added. Instead we compute the value of &#916;U if a particle were inserted.</p>

<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.widom.WidomApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p class="header_title"><b>Algorithm</b></p>

<ol>

<li>Use the Metropolis (or other) algorithm to simulate a system of particles at fixed temperature, volume, and number of particles. In our program  we simulate a system of Lennard-Jones particles in two dimensions.</li>

<li>Periodically add a "ghost" particle at a random position and compute the change in energy &#916;U that would result. In our simulation we do such a trial  after each MC step.</li>

<li>Compute the average of exp(-&#916;U/kT) and estimate &#956;* = -kT&lt;exp(-&#916;U/kT)&gt;.</li>

</ol>

<p class="header_title">Problems</p>

<ol>

<li>Run the simulation using the default parameters. After the average value for the mean excess chemical potential has reached an apparent equilibrium value, press <tt>Reset Data</tt>, and run  until &#956;* has reached an approximately constant value with small fluctuations. Record your value for  &#956;*. Is it positive or negative? Explain why it has the sign that it does. Remember that &#956;* is  in addition to that of an ideal gas at the same temperature.</li>

<li>Repeat your simulations for different temperatures. Does the excess chemical potential increase or decrease with temperature? Does it change sign? How does the dependence of &#956;* on temperature compare with the dependence of the potential energy on the temperature?</li>

<li>Repeat your simulations for T = 1 but at different densities. For example try 30 &#215; 30 and 20  &#215; 20 cells. How does &#956;* depend on the density?  Compare its dependence to that of the mean potential energy.</li>

</ol>

<p class="header_title">References</p>

<ul>

<li>B. Widom, "Some topics in the theory of fluids," J. Chem. Phys. <b>39</b>, 2802-2812 (1963).</li>

<li>Daan Frenkel and Berend Smit, <i>Understanding Molecular Simulation</i> (Academic Press, San Diego, 1996), pp. 157-160.</li>

</ul>

<p class="header_title">Java Classes</p>

<ul>

<li>WidomApp</li>
<li>Widom</li>
<li>LJParticlesWidomLoader</li>

</ul>

<p class = "small">Updated 3 May 2007.</p>

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